Charles De Clercq: Classification of upper motives of algebraic groups of type A_n

Submission: 2011, Jan 25

Let A, A' be two central simple algebras over a field K and F be a finite field of characteristic p. We prove that the upper indecomposable direct summands of the motives of two anisotropic varieties of flags of right ideals X(d_1,...,d_k;A) and X(d'_1,...,d'_s;A') with coefficients in F are isomorphic if and only if the p-adic valuations of gcd(d_1,...,d_k) and gcd(d'_1,..,d'_s) are equal and the classes of the p-primary components A_p and A'_p of A and A' generate the same group in the Brauer group of K. This result leads to a surprising dichotomy between upper motives of absolutely simple adjoint algebraic groups of inner type A_n.

2010 Mathematics Subject Classification: 20G15,14M15,14M17

Keywords and Phrases: algebraic groups, projective homogeneous varieties, motivic decompositions

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